Unsolved Problems in Combinatorial Games

Richard K. Guy & Richard J. Nowakowski

2008-02-24

We have sorted the problems into sections:

A. Taking & Breaking • B. Pushing & Placing Pieces • C. Playing with Pencil & Paper • D. Disturbing & Destroying • E. Theory of Games •

They have been given new numbers. The numbers in parentheses are the old numbers used in each of the lists of unsolved problems given on pp. 183–189 of AMS Proc. Sympos. Appl. Math. 43(1991), called PSAM 43 below; on pp. 475–491 of Games of No Chance, hereafter referred to as GONC; and on pp. 457–473 of More Games of No Chance (MGONC). Missing numbers are of problems which have been solved, or for which we have nothing new to add. References [year] may be found in Fraenkel’s Bibliography at the end of this volume. References [#] are at the end of this article. A useful reference for the rules and an introduction to many of the speciﬁc games mentioned below is M. Albert, R. J. Nowakowski & D. Wolfe, Lessons in Play: An Introduction to the Combinatorial Theory of Games, AKPeters, 2007 (LIP).

1 A. Taking & Breaking Games

A1(1). Subtraction games with ﬁnite subtraction sets are known to have periodic nim-sequences. Investigate the relationship between the subtraction set and the length and structure of the period. The same question can be asked about partizan subtraction games, in which each player is assigned an individual subtraction set. See Fraenkel & Kotzig [1987].

[A move in the game S(s1,s2,s3,...) is to take a number of beans from a heap, provided that number is a member of the subtraction-set, s1,s2,s3,... . Analysis of such a game and of many other heap games is conveniently{ recorded} by a nim-sequence, n0n1n2n3 ..., meaning that the nim-value of a heap of h beans is nh; i.e., that the value of a heap of h beans in this particular game is the nimber n .] ∗ h For examples see Table 2 in 4 on p. 67 of the Impartial Games paper in GONC. § It would now seem feasible to give the complete analysis for games whose subtraction sets have just three members, though this has so far eluded us. Several people, including Mark Paulhus and Alex Fink, have given a complete analysis for all sets 1, b, c and for sets a, b, c with a < b < c < 32. { } { } In general, period lengths can be surprisingly long, and it has been suggested that they could be super-polynomial in terms of the size of the subtraction set. However, Guy conjectures that they are bounded by polynomials of n degree at most 2 in sn, the largest member of a subtraction set of cardinality n. It would also be of interest to characterize the subtraction sets which yield ! " a purely periodic nim-sequence, i.e., for which there is no preperiod. Angela Siegel [18] considered infinite subtraction sets which are the comple- ment of finite ones and showed that the nim-sequences are always arithmetic periodic. That is, the nim-values belong to a finite set of arithmetic progressions with the same common difference. The number of progressions is the period and their common difference is called the saltus. For instance, the game S 4ˆ, 9ˆ, 26, 30 (in which a player may take any number of beans except 4, 9, 26{ or 30) has} a pre-period of length 243, period-length 13014 and saltus 4702. # # For infinite subtraction games in general there are corresponding questions about the length and purity of the period.

2 As we go to press we note that Question A2 on the 2006-12-02 Putnam exam is the subtraction game with subtraction set p 1:p prime . Show { − } that there are inﬁnitely many heap sizes which are -positions. P A2(2). Are all ﬁnite octal games ultimately periodic? [If the binary expansion of the kth code digit in the game with code d0 d1d2d3 ... is · ak bk ck dk =2 +2 +2 + ..., where 0 a

1024–

512–

256–

Figure 1: Plot of 11000000 nim-values of the octal game 007 ·

3 Perhaps the most notorious and deserving of attention is the game 007, one-dimensional Tic-Tac-Toe, or Treblecross, which Flammenkamp has pushed· to 225. Figure 1 shows the first 11 million nim-values, a small proportion of which are 1024; the largest, (6193903) = 1401 is shown circled. Will 2048 ever be reached?≥ G Achim Flammenkamp has settled 106: it has the remarkable period and preperiod lengths of 328226140474 and· 465384263797. For information on the current status of each of these games, see Flammenkamp’s web page at http://www.uni-bielefeld.de/~achim/octal.html A game similar to Grundy’s, and which is also unsolved, is John Conway’s Couples-Are-Forever (LIP, p. 142) where a move is to split any heap except a heap of two. The first 50 million nim-values haven’t displayed any periodicity. See Caines et al. [1999]. More generally, Bill Pulleyblank suggests looking at splitting games in which you may only split heaps of size >h, so that h =1 is She-Loves-Me-She-Loves-Me-Not and h = 2 is Couples-Are-Forever. David Singmaster suggested a similar generalization: you may split a heap provided that the resulting two heaps each contain at least k beans: k = 1 is the same as h = 1, while k = 2 is the third cousin of Dawson’s Chess. Explain the structure of the periods of games known to be periodic. In Discrete Math., 44(1983) 331–334, Problem 38, Fraenkel raised questions concerning the computational complexity (see E1 below) of octal games. In Problem 39, he & Kotzig define partizan octal games in which distinct octals are assigned to the two players. Mesdal, in this volume, pp.??, show that in many cases, if the game is “all-small” (WW, pp. 229–262, LIP, pp. 183–207), then the atomic weights are arithmetic periodic. In Problem 40, Fraenkel introduces poset games, played on a partially ordered set of heaps, each player in turn selecting a heap and then removing a non-negative number of beans from this heap and from each heap above it in the ordering, at least one heap being reduced in size. For posets of height one, new regularities in the nim-sequence can occur; see Horrocks & Nowakowski [2003]. Note that this includes, as particular cases, Subset Takeaway, Chomp or Divisors, and Green Hackenbush forests. Compare Problems A3, D1 and D2 below.

A3(3). Hexadecimal games have code digits dk in the interval from 0 to f (= 15), so that there are options splitting a heap into three heaps. See WW, 116–117.

4 Such games may be arithmetically periodic. Nowakowski has calculated the first 100000 nim-values for each of the 1-, 2- and 3-digit games. Richard Austin’s theorem 6.8 in his thesis [1976] and the generalization by Howse & Nowakowski [2004] suffice to confirm the arithmetic periodicity of several of these games. Some interesting specimens are 28 = 29, which have period 53 and saltus 16, the only exceptional value being· (0) =· 0; 9c, which has period 36, preperiod 28 and saltus 16; and f6 withG period 43· and saltus 32, but its apparent · preperiod of 604 and failure to satisfy one of the conditions of the theorem prevent us from verifying the ultimate periodicity. The game 205200c is · arithmetic periodic with pre-period length of 4, period length of 40, saltus 16 except that 40k +19 has nim-value 6 and 40k +39 has nim-value 14. This regularity, (which also seems to be exhibited by 660060008 with a period length of · approximately 300,000), was first reported in Horrocks & Nowakowski [2003] (see Problem A2.) Grossman & Nowakowski [7] have shown that the nim- sequences for 200. . . 0048, with an odd number of zero code digits, exhibit · “ruler function” patterns. The game 9 has not so far yielded its complete analysis, but, as far as analyzed (to heaps· of size 100000), exhibits a remarkable fractal-like set of nim-values. See Howse & Nowakowski [2004]. Also of special interest are e; 7f (which has a strong tendency to period 8, saltus 4, but, for n 100, 000,· has· 14 exceptional values, the largest being (94156) = 26614); ≤ G b6 (which “looks octal”); b33b (where a heap of size n has nim-value n except· for 27 heap sizes which· appear to be random); and 817264517 · [why 817264517 ?] whose nim-values appear to form a lattice of ruler functions 1 1 with slopes slightly less than 2 and 2 (see Figure 2). The largest value in the range calculated is (206265) = 101458.− G Other unsolved hexadecimal games are 1x with x 8, 9, c, d, e, f ; · ∈{ } 2x, a x f; 3x,8 x e; 4x, x 9, b, d, f ; 5x,8 x f; ·6x,8≤ x ≤ f; ·7x,8≤ x ≤ f; ·9x,1∈{x a; 9d} ;· bx, x≤ 6≤, 9,d ; ·dx,1≤ x ≤ f;· and fx≤with≤ x · 4, 6,≤7 . ≤ · · ∈{ } · ≤ ≤ · ∈{ }

A4(53). N-heap WythoffGame. Given N 2 heaps of finitely ≥ many tokens, whose sizes are p1, . . . , pN with p1 pN . Players take turns removing any positive number of tokens from≤ a ···single≤ heap or removing (a1, . . . , aN ) from all the heaps — ai from the i-th heap — subject to the con- N ditions: (i) 0 ai pi for each i, (ii) i=1 ai > 0, (iii) a1 aN = 0, where is nim addition.≤ ≤ The player making the last move wins⊕··· and⊕ the opponent ⊕ loses. Note that the classical Wythoff$game is the case N = 2.

5 For N 3, Fraenkel makes the following conjectures. ≥ 1 N 2 Conjecture 1. For every ﬁxed set K := (A ,...,A − ) there exists an integer m = m(K) (i.e, m depends only on K), such that

1 N 2 N 1 N N 2 N 1 N (A ,...,A − ,A − ,A ),A− A − A n n ≤ n ≤ n N 1 N 1 with A −

x K : 0 k < K s.t. (k, K, x) is a P position 0,...,K 1 { ≥ ∃ ≤ − }∪{ − } The following upper bound for A3 was established: A3 (K + 3)A2 +2K + 2. n n ≤ n It was also proved that Conjecture 1 implies Conjecture 2. In Sun & Zeilberger [2004], a sufficient condition for the conjectures to hold was given. It was then proved that the conjectures are true for the case N = 3, where the first heap has up to 10 tokens. For those 10 cases, the parameter values m, M, a, T were listed in a table. Sun [2005] obtained results similar to those in Fraenkel & Krieger [2004], but the proofs are different. It was also proved that Conjecture 1 implies

6 Conjecture 2. A method was given to compute a in terms of certain indexes of the Ai and Bj.

A5(23). Burning-the-Candle-at-Both-Ends. Conway and Fraenkel ask us to analyze Nim played with a row of heaps. A move may only be made in the leftmost or in the rightmost heap. When a heap becomes empty, then its neighbor becomes the end heap. Albert & Nowakowski [2001] have determined the outcome classes in impartial and partizan versions (called End-Nim, LIP, pp. 210, 263) with finite heaps, and Duffy, Kolpin & Wolfe, in this volume, pp.??, extend the partizan case to infinite ordinal heaps. Wolfe asks for the actual values. Nowakowski suggested to analyze impartial and partizan End-Wythoff: take from either end-pile, or the same number from both ends. The impartial game is solved by Fraenkel & Reisner, in this volume, pp.??. Fraenkel [1982] asks a similar question about a generalized Wythoffgame: take from either end-pile or take k>0 from one end-pile and $> 0 from the other, subject to k $

7 The partizan game of Red-Blue Cherries is played on an arbitrary graph. A player picks an appropriately colored cherry from a vertex of minimum degree, which disappears at the same time. Albert et al.[1] show that if the graph has a leaf, then the value is an integer. See also McCurdy [10].

A6(17). Extend the analysis of Kotzig’s Nim (WW, 515–517). Is the game eventually periodic in terms of the length of the circle for every ﬁnite move set? Analyze the mis`ere version of Kotzig’s Nim.

A7(18). Obtain asymptotic estimates for the proportions of -, - and N O -positions in Epstein’s Put-or-Take-a-Square game (WW, 518–520). P A8. Gale’s Nim. This is Nim played with four heaps, but the game ends when three of the heaps have vanished, so that there is a single heap left. Brouwer and Guy have independently given a partial analysis, but the situation where the four heaps have distinct sizes greater than 2 is open. An obvious generalization is to play with h heaps and play ﬁnishes when k of them have vanished.

A9. Euclid’s Nim is played with a pair of positive integers, a move being to diminish the larger by any multiple of the smaller. The winner is the player who reduces a number to zero. Analyses have been given by Cole & Davie [1969], Spitznagel [1973], Lengyel [2003], Collins [2005], Fraenkel [2005] and Nivasch [2006]. Gurvich [8] shows that the nim-value, g+(a, b) for the pair (a, b) in normal play is the same as the misère nim-value, g−(a, b) except for (a, b)=(kFi, kFi+1) where k>0 and Fi is the i-th Fibonacci number. In this + case, g (kFi, kFi+1) = 0 and g−(kFi, kFi+1) = 1 if i is even and the values are reversed if i is odd. We are not aware of an analysis of the game played with three or more integers. A10(20). Some advance in the analysis of D.U.D.E.N.E.Y (WW, 521– 523) has been made by Marc Wallace, Alex Fink and Kevin Saff. [The game is Nim, but with an upper bound, Y , on the number of beans that may be taken, and with the restriction that a player may not repeat his opponent’s last move. If Y is even, the analysis is easy.] We can, for example, extend the table of strings of pearls given in WW, p.523, with the following values of Y which have the pure periods shown, where D=Y + 2, E=Y + 1. The first entry corrects an error of 128r + 31 in WW.

8 256r + 31 DEE 512r + 153 DEE 1024r + 415 DEE 512r + 97 DDEDDDE 512r + 159 DEE 512r + 425 DE 1024r + 103 DE 512r + 225 DDE 512r + 487 DEE 128r + 119 DEE 512r + 255 E 1024r + 521 DDDE 1024r + 127 DEEE 512r + 257 DDDDE 1024r + 607 DDE 512r + 151 DDDEE 512r + 297 DDEDEDE 1024r + 735 DEEE

It seems likely that the string for Y =22k+1 +22k 1 has the simple period − E for all values of k. But the following evidence of the fraction, among 2k cases, that remain undetermined:

k = 3567 8 9 10 11 12 13 14 15 16 17 1 5 9 11 21 33 60 97 177 304 556 974 1576 2763 fraction 2 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 65536 suggests that an analysis will never be complete. Moreover, the periods of the pearl-strings appear to become arbitrarily long.

A11(21). Schuhstrings is the same as D.U.D.E.N.E.Y, except that a deduction of zero is also allowed, but cannot be immediately repeated (WW, 523–524). In Winning Ways it was stated that it was not known whether there is any Schuhstring game in which three or more strings terminate simultaneously. Kevin Saﬀhas found three such strings (when the maximum deduction is Y = 3430, the three strings of multiples of 2793, 3059, 3381 terminate simultaneously) and he conjectures that there can be arbitrarily many such simultaneous terminations. A12(22). Analyze Dude, i.e., unbounded Dudeney, or Nim in which you are not allowed to repeat your opponent’s last move. Let [h ,h , . . . , h ; m], h h , be the game with heaps of size h through 1 2 k i ≤ i+1 1 hk, where m is the move just made and m = 0 denotes a starting position. Then [4], for k = 1 the -positions are [(2s + 1)22j; (2s + 1)22j]; for k = 2 they P are [(2s + 1)22j, (2s + 1)22j; 1]; and for k 3 the heap sizes are arbitrary, the only condition being that the previous move≥ was 1. The nim-values do not seem to show an easily described pattern.

A13. Nim with pass. David Gale would like to see an analysis of Nim played with the option of a single pass by either of the players, which may be made at any time up to the penultimate move. It may not be made at the end of the game. Once a player has passed, the game is as in ordinary Nim. The game ends when all heaps have vanished.

9 A14. Games with a Muller twist. In such games, each player speciﬁes a condition on the set of options available to her opponent on his next move. In Odd-or-Even Nim, for example, each player speciﬁes the parity of the opponent’s next move. This game was analyzed by Smith & St˘anic˘a [2002], who propose several other such games which are still open (see also Gavel & Strimling [2004]). The game of Blocking Nim proceeds in exactly the same way as ordinary Nim with N heaps, except that before a given player takes his turn, his opponent is allowed to announce a block,(a1, . . . aN ); i.e., for each pile of counters, he has the option of specifying a positive number of counters which may not be removed from that pile. Flammenkamp, Holshouser & Reiter [2003, 2004] give the -positions for three-heap Blocking Nim with an incomplete block containingP only one number, and ask for an analysis of this game with a block on just two heaps, or on all three. There are corresponding questions for games with more than three heaps.

A15(13). Misère analysis has been revolutionized by Thane Plambeck and Aaron Siegel with their concept of the misère quotient of a game [13], though the number of unsolved problems continues to increase. Let be some set of games played under misère rules. Typically, is the set of positionsA that arise in a particular game, such as Dawson’s Chess.A Games H,K are said to be equivalent, denoted by H K, if H + X and K + X ∈A ≡ have the same outcome for all games X . The relation is an equivalence relation, and a set of representatives, one∈A from each equivalence≡ class, forms the misère quotient, = / .Aquotient map Φ: is defined, for G , by Φ: =[G] Q. A ≡ A→Q ∈A G ≡ Plambeck and Siegel ask the specific questions: (1) The misère quotient of 07 (Dawson’s Kayles) has order 638 at heap · size 33. Is it infinite at heap size 34? Even if the misère quotient is infinite at heap 34 then, by Redei’s theorem [6, p. 142], [14, p. ], it must be isomorphic to a finitely-presented commutative monoid. Call this monoid D34. Exhibit a monoid presentation of D34, and having done that, exhibit D35, D36, etc, and explain what is going on in general. Given a set of games , describe A an algorithm to determine whether the misère quotient of is infinite. Much harder: if the quotient is infinite, give an algorithm to computeA a presentation for it.

10 (2) A quotient map Φ: is said to be faithful if, whenever Φ(G)= Φ(H), then G and H haveA the→Q same normal-play Grundy value. Is every quotient map faithful? (3) Let ( , ) be a quotient and a maximal subgroup of . Must be nonempty?Q P (Note: it’s easy to getS a “yes” answer in the specialQ caseS when∩P is the kernel) S (4) Give complete misère analyses for any of the (normal-play periodic) octal games that show “algebraic-periodicity” in misère play. Some examples are 54, 261, 355, 357, 516 and 724. Give a precise definition of algebraic periodicity· · and· describe· an· algorithm· for detecting and generalizing it. This is a huge question: if such an algorithm exists, it would likely instantly give solutions to at least a half-dozen unsolved 2- and 3-digit octals. (5) Extend the classification of misère quotients. We have preliminary results on the number of quotients of order n 18 but believe that this can be ≤ pushed far higher. (6) Exhibit a misère quotient with a period-5 element. Same question for period 8, etc. We’ve detected quotients with elements of periods 1, 2, 3, 4, 6, and infinity, and we conjecture that there is no restriction on the periods of quotient elements. (7) In the flavor of both (5) and (6): What is the smallest quotient containing a period 4 (or 3 or 6) element? Plambeck also offers prizes of US$500.00 for a complete analysis of Dawson’s Chess, 137 (alias Dawson’s Kayles, 07); US$200.00 for the “wild quaternary · · game”, 3102; and US$25.00 each for 3122, 3123 and 3312. · · · · There is a website http://www.miseregames.org which contains thou- sands of misère quotients for octal games. Siegel notes that Dawson first proposed his problem in 1935, making it perhaps the oldest open problem in combinatorial game theory. [Michael Albert offers the alternative “Is chess a first player win?”] It may be of historical interest to note that Dawson showed the problem to one of the present authors around 1947. Fortunately, he forgot that Dawson proposed it as a losing game, was able to analyze the normal play version, rediscover the Sprague-Grundy theory, and get Conway interested in games.

11 B. Pushing & Placing Pieces

B1(5). The game of Go is of particular interest, partly because of the loopiness induced by the “ko” rule, and many problems involve general theory: see E4 and E5.

13 12 11 10 9 8 7 6 5 4 3 2 1 ABCDEFGHJKLMN ABCDEFGHJKLMN Figure 2: Jiang v. Rui, MSRI, July, 2000

Elwyn writes: I attach one region that has been studied intermittently over the past several years. The region occurs in the southeast corner of the board (Figure 3). At move 85 Black takes the ko at L6. What then is the temperature at N4 ? This position is copied from the game Jiang and Rui played at MSRI in July 2000. In 2001, Bill Spight and I worked out a purported solution by hand, assumingeither Black komaster or White komaster. I’ve recently been trying to get that rather complicated solution conﬁrmed by GoExplorer, which would then presumably also be able to calculate the dogmatic solution. I’ve been actively pursuing this oﬀand on for the past couple weeks, and haven’t gotten there yet. Elwyn also writes: Nakamura has shown (this volume, pp.??) how capturing races in Go can be analyzed by treating liberties as combinatorial games. Like atomic weights, when the values are integers, each player’s best move reduces his opponent’s resources by one. The similarities between atomic weights and Nakamura’s liberties are striking.

12 Theoretical problem: Either find a common formulation which includes much or all of atomic weight theory and Nakamura’s theory of liberties, OR find some significant differences. Important practical applied problem: Extend Nakamura’s theory to include other complications which often arise in Go, such as simple kos, either internal and/or external.

B2. A simpler game involving kos is Woodpush (see LIP, pp. 214, 275). This is played on a finite strip of squares. Each square is empty or occupied by a black or white piece. A piece of the current player’s color retreats: Left retreats to the left and Right to the right—to the next empty square, or offthe board if there is no empty square; except, if there is a contiguous string containing an opponent’s piece then it can move in the opposite direction pushing the string ahead of it. Pieces can be moved offthe end of the strip. Immediate repetition of a global board positions is not allowed. A “ko” threat must be played first. For example

Left Right Left Right LRR LRR LR R ko-threat R R ! → ! → ! → → !! Note that Right’s ﬁrst move to LRR! is illegal because it repeats the immediately prior board position and Left’s second move to !LRR is also illegal so he must play a ko-threat. Also note that in !LRR!, Right never has to play a ko-threat since he can always push with either of his two pieces—with Left moving ﬁrst,

Left Right !LRR! !!LRR !LR!R → ko-threat → Right answers ko-threat → LRR → LRR → !! → ! ! Berlekamp, Plambeck, Ottaway, Aaron Siegel & Spight (work in progress) use top-down thermography to analyze the three piece positions. What about more pieces?

B3(40). Chess. Noam Elkies [2002] has examined Dawson’s Chess, but played under usual Chess rules, so that capture is not obligatory. He would still welcome progress with his conjecture that the value k occurs ∗ for all k in (ordinary Chess) pawn endings on suﬃciently large chessboards.

13 Thea van Roode has suggested Impartial Chess, in which the players may make moves of either color. Checks need not be responded to and Kings may be captured. The winner could be the ﬁrst to promote a pawn.

B4(30). Low & Stamp [2006] have given a strategy in which White wins the King and Rook vs. King problem within an 11 9 region. ×

B5. Non-attacking Queens. Noon & Van Brummelen [2006] alternately place queens on an n n chessboard so that no queen attacks another. The winner is the last queen× placer. They give nim-values for boards of sizes 1 ≤ n 10 as 1121312310 and ask for the values of larger boards. ≤

B6(55). Amazons. Martin M¨uller [11] has shown that the 5 5 game is a ﬁrst player win and asks about the 6 6 game. × ×

B7. Conway’s Philosopher’s Football, or Phutball, is usually played on a Go board with positions (i, j), 9 i, j 9 and the ball starting at (0,0). − ≤ ≤ For the rules, see WW, pp.752–755. The game is loopy (see E5 below), and Nowakowski, Ottaway & Siegel (see [17]) discovered positions that contained tame cycles, i.e., cycles with only two strings, one each of Left and Right moves. Aaron Siegel asks if there are positions in such combinatorial games which are stoppers but contain a wild cycle, i.e., one which contains more than one alternation between Left and Right moves. Demaine, Demaine & Eppstein [2002] show that it is NP-complete to decide if a player can win on the next move. Phlag Phutball is a variant played on an n n board with the initial position of the ball at (0, 0) except that now only× the ball may occupy the positions (2i, 2j) with both coordinates even. This eliminates “tackling”, and is an extension of 1-dimensional Oddish Phutball, analyzed in Grossman & Nowakowski [2002]. The (3, 2n + 1) board (i.e. (i, j), i =0, 1, 2 and n j n) is already interesting and requires a different strategy from that appropriate− ≤ ≤ to Oddish Phutball. B8. Hex. (LIP, pp. 264–265) Nash’s strategy stealing argument shows that Hex is a first player win but few quantitative results are known. Garikai Campbell [2004] asks (1) For each n, what is the shortest path on an n n board with which the first player can guarantee a win? ×

14 (2) What is the least number of moves in which the first player can guarantee a win? B9(54). Fox and Geese. Berlekamp & Siegel [17, Chapter 2] and WW pp.669–710, “analysed the game fairly completely, relying in part on results obtained using CGSuite.” On p.710 of WW the following open problems are given. 1. Define a position’s span as the maximum occupied row-rank minus its minimum occupied row-rank. Then quantify and prove an assertion such as the following: If the backfield is sufficiently large, and the span is sufficiently large, and if the separation is sufficiently small, and if the Fox is neither already trapped in a daggered position along the side of the board, nor immediately about to be so trapped, then the Fox can escape and the value is off. 2. Show that any formation of three Geese near the centre of a very tall board has a “critical rank” with the following property: If the northern Goose is far above, and the Fox is far below, then the value of the position is either positive, HOT, or off, according as the northern Goose is closer, equidistant, or further from the critical rank than the Fox. 3. Welton asks what happens if the Fox is empowered to retreat like a Bishop, going back several squares at a time in a straight line ? More generally, suppose his straightline retreating moves are confined to some specific set of sizes. Does 1,3 , which maintains parity, give him more or less advantage than 1,2 ? { } { } 4. What happens if the number of Geese and boardwidths are changed ? In Aaron Siegel’s thesis there are several other questions: 5. In the critical position, with Geese at [we use the algebraic Chess nota- tion of a, b, c, d, ...for the files and 1, 2, 3, ..., n for the ranks] (b,n), (d,n), (n 8) (e,n 1), (g,n 1), and Fox at (c,n 1), which has value 1 + 2− − on an n 8 board− with n− 8 in the usual game,− is the value 2n+11 for all n 6 when× played with “Ceylonese≥ rules” ? (Fox allowed two− moves at each turn.)≥ 6. On an n 4 board with n 5 and Geese at (b,n) and (c,n 1) do all Fox positions have× value over ? With≥ the Geese on (b,n) and (d,n−) are only other values 0 at (c,n 1) and over 0 at (b,n 2) and (d,n 2) ? − { | } − − 7. On an n 6 board with n 8 and Geese at (b,n), (d,n) and (e,n 1) × ≥ − do the positions (a,n 2k+1), (c,n 2k+1), (e,n 2k+1), all have value 0, and those at (b,n 2k), (d,−n 2k), (f,n−2k) all have− value Star?˙ And if the Geese − − − are at (b,n), (d,n) and (f,n) are the zeroes and Stars interchanged ?

15 B10. Hare and Hounds.

Figure 3: Sequences of Hare and Hounds positions

Aaron Siegel asks if the sequences of positions of increasing board length shown in Figure 4 are, on the left, increasingly hot, and, on the right, have arbitrarily large negative atomic weight. He also conjectures that the starting position on a 6n+5 3 board, for n>0, has value ×

(n 1) + b, c 0 0 0 0 ... 0 − − | 2 ' % & & & & & & & & where there are 2n + 4 zeroes and slashes and b = &0,a 0, 0 oﬀ , { | } ( & ) c = 0 2 0 0 and a = 0, 2 0, 2 & ↓→ ∗| { ↓→ ∗| ↓→ ∗} & % & * & & & &

16 B11(4). Extend the analysis of Domineering (WW, pp. 119–122, 138– 142; LIP pp. 1–7, 260). [Left & Right take turns to place dominoes on a checker-board. Left orients her dominoes North-South and Right orients his East-West. Each domino exactly covers two squares of the board and no two dominoes overlap. A player unable to play loses.] See Berlekamp [1988] and the second edition of WW, 138–142, where some new values are given. For example David Wolfe & Dan Calistrate have found the values (to within ‘-ish’, i.e., infinitesimally shifted) of 4 8, 5 6 and 6 6 boards. The value for a 5 7 board is × × × × 3 5 1 3 1 3 3 1 , , 1 1 3 1, 1 3 +2 4 − 2 2 − 2 2 − − − & − 2 − 2 − − ' , % , * % , % , * & , * & % , & * , , , , , & , & , & , , , , , & , & , & , Lachmann,, Moore &, Rapaport [2002]& discover who& wins on rectangular,, & toroidal, and cylindrical boards of widths 2, 3, 5 and 7, but do not find their values. Bullock [3, p. 84] showed that 19 4, 21 4, 14 6 and 10 8 are wins for Left and that 10 10 is a first player× win.× × × × Berlekamp notes that the value of a 2 n board, for n even, is only known to within“ish”, and that there are problems× on 3 n and 4 n boards that × × are still open. Berlekamp asks, as a hard problem, to characterize all hot Domineering positions to within “ish”. As a possibly easier problem he asks for a Domineering position with a new temperature, i.e., one not occurring in Table 1 on GONC, p. 477. Gabriel Drummond-Cole (2002) found values with temperatures between 1.5 and 2. Figure 5 shows a position of value 2 and temperature 2. ± ∗

         R   L   

          Figure 4: A Domineering position of value 2  ± ∗

17 Shankar & Sridharan [2005] have found many Domineering positions with temperatures other than those shown in Table 1 on p.477 of GONC. Blanco & Fraenkel [2] have obtained partial results for the game of Tromineering, played with trominoes in place of (or, alternatively, in addition to) dominoes.

C. Playing with Pencil & Paper

C1(51). Elwyn Berlekamp asks for a complete theory of “Icelandic” 1 n Dots-and-Boxes, i.e., with starting position as in Figure 6. ×

Figure 5: Starting position for “Icelandic” 1 n Dots-and-Boxes × See Berlekamp’s book [2000] for more problems about this popular chil- dren’s (and adults’) game and see also WW, pp. 541–584; LIP, pp. 21–28, 260.

C2(25). Extend the analysis of the Conway-Paterson game of Sprouts in either the normal or mis`ere form. (WW, pp. 564–568). [A move joins two spots, or a spot to itself by a curve which doesn’t meet any other spot or previously drawn curve. When a curve is drawn, a new spot must be placed on it. The valence of any spot must not exceed three.]

C3(26). Extend the analysis of Sylver Coinage (WW, 575–597). [Players alternately name diﬀerent positive integers, but may not name a number which is the sum of previously named ones, with repetitions allowed. Whoever names 1 loses.] Sicherman [2002] contains recent information.

C4(28). Extend Ulehla’s´ or Berlekamp’s analysis of von Neumann’s game from directed forests to directed acyclic graphs (WW, 570–572; Ulehla´ [1980]). [von Neumann’s game, or Hackendot, is played on one or more rooted trees. The roots induce a direction, towards the root, on each edge. A move is to delete a node, together with all nodes on the path to the root, and all edges

18 incident with those nodes. Any remaining subtrees are rooted by the nodes that were adjacent to deleted nodes.] C5(43). Inverting Hackenbush. Thea van Roode has written a thesis [15] investigating both this and Reversing Hackenbush, but there is plenty of room for further analysis of both games. In Inverting Hackenbush, when a player deletes an edge from a component, the remainder of the component is replanted with the new root being the pruning point of the deleted edge. In Reversing Hackenbush, the colors of the edges are all changed after each deletion. Both games are hot, in contrast to Blue-Red Hackenbush (WW, pp. 1–7; LIP, pp. 82, 88, 111–112, 212, 266) which is cold, and Green Hackenbush (WW, pp. 189–196), which is tepid. C6(42). Beanstalk and Beans-Don’t-Talk are games invented respec- tively by John Isbell and John Conway. See Guy [1986]. Beanstalk is played between Jack and the Giant. The Giant chooses a positive integer, n0. Then J. and G. play alternately n1, n2, n3, ...according to the rule ni+1 = ni/2 if ni is even, = 3ni 1 if ni is odd; i.e. if ni is even, there’s only one option, while if ni is odd there± are just two. The winner is the person moving to 1. We still don’t know if there are any -positions (positions of inﬁnite re- moteness). O

C7(63). The Erd˝os-Szekeres game [5] (and see Schensted [16]) was introduced by Harary, Sagan & West [1985]. From a deck of cards labelled from 1 through n, Alexander and Bridget alternately choose a card and append it to a sequence of cards. The game ends when there is an ascending subsequence of a cards or a descending subsequence of d cards. The game appears to have a strong bias towards the first player. Albert et al., this volume, pp.??, show that for d = 2 and a n the outcome is or according as n is odd or even, and is (drawn) if n≤ < a. They conjectureN thatP O for a d 3 and all sufficiently large n, it is with both normal and misère play,≥ and≥ also with normal play when played withN the rationals in place of the first n integers. They also suggest investigating the form of the game in which players take turns naming pairs (i, πi) subject to the constraint that the chosen values form part of the graph of some permutation of 1, 2, . . . , n . { }

19 D. Disturbing & Destroying

D1(27). Extend the analysis of Chomp (WW, 598–599, LIP 19, 46, 216). David Gale offers $300.00 for the solution of the infinite 3-D version where the board is the set of all triples (x, y, z) of non-negative integers, that is, the lattice points in the positive octant of R3. The problem is to decide whether it is a win for the first or second player. Chomp (Gale [1974]) is equivalent to Divisors (Schuh [1952]). Chomp is easily solved for 2 n arrays, Sun [2002], and indeed a recent result by Steven Byrnes [2003] shows× that any poset game eventually displays periodic behavior if it has two rows, and a fixed finite number of other elements. See also the Fraenkel poset games mentioned near the end of A2. Thus, most of the work in recent years has been on three-rowed Chomp. The situation becomes quite complicated when a third row is added, see Zeil- berger [2001] and Brouwer et al. [2005]. A novel approach (renormalization) is taken by Friedman & Landsberg (see this volume, pp.??). They demonstrate that three-rowed Chomp exhibits certain scaling and self-similarity patterns similar to chaotic systems. Is there a deterministic proof that there is a unique winning move from a 3 n rectangle? The renormalization approach is based × on statistical methods and has caused some controversy. In fact, Fraenkel has written

Ivars Peterson [12] introduced a physics-based approach for the theory of combinatorial games, due to professors Eric Friedman and Adam Landsberg . . . . Throughout their and Peterson’s articles, it’s unclear whether their claims have been rigorously proved or are the result of “curve ﬁtting” the experimental data that has been computed by mathematicians working on Chomp. Over the time— including their presentation at [this] workshop in June 2005—it became apparent that it’s the latter. . . . there is no place for claims such as “breakthrough” for Fried- man and Landsberg’s approach, nor for creating or even permitting fuzziness between experimental data and proven results. However, their approach may lead to results in the future, and is therefore of potential promise.

Transfinite Chomp has been investigated by Huddleston & Shurman [2002]. An open question is to calculate the nim value of the position ω 4 × 20 —they conjecture this to be ω 2, but it could be as low as 46, or even un- computable! Perhaps the most fascinati· ng open question in Transfinite Chomp is their Stratification Conjecture, which states that if the number of elements taken in a move is <ωi, then the change in the nim-value is also <ωi.

D2(33). Subset Take-away. Given a finite set, players alternately choose proper subsets subject to the rule that once a subset has been chosen no proper subset can be removed. Last player to move wins. Many people play the dual, i.e. a non-empty subset must be chosen and no proper superset of this can be chosen. We discuss this version of the game which now can be considered a poset game with the sets ordered by inclusion. The (n; k) Subset Take-away game is played using all subsets of sizes 1 through k of a n-element set. In the (n; n) game one has the whole set (i.e. the set of size n) as an option, so a strategy-stealing argument shows this must be a first player win. 1. Gale & Neyman [1982], in their original paper on the game, conjectured that the winning move in the (n; n) game is to remove just the whole set. This is equivalent to the statement that the (n; n 1) game is a second-player win, which has been verified only for n 5. − ≤ 2. A stronger conjecture states that (n; k) is a second player win if and only if k + 1 divides n. This was proved in the original paper only for k = 1 or 2. See also Fraenkel & Scheinerman [1991].

D3(39). Sowing or Mancala games. There appears to have been no advance on the papers mentioned in MGONC, although we feel that this should be a fruitful ﬁeld of investigation at several diﬀerent levels.

D4. Annihilation games. k-Annihilation. Initially place tokens on some of the vertices of a ﬁnite digraph. Denote by ρout(u) the outvalence of a vertex u. A move consists of removing a token from some vertex u, and “complementing” t := min(k,ρout(u)) (immediate) followers of u, say v1, . . . , vt: if there is a token on vj, remove it; if there is no token there, put one on it. The player making the last move wins. If there is no last move, the outcome is a draw. For k = 1, there is an O(n6) algorithm for deciding whether any given position is in , , or ; and for computing an optimal next move in the P N O last 2 cases (Fraenkel & Yesha [1982]). Fraenkel asks: Is there a polynomial algorithm for k>1? For an application of k-annihilation games to lexicodes, see Fraenkel & Rahat [2003].

21 D5. Toppling dominoes (LIP, pp. 110–112, 274) is played with a row of vertical dominoes each of which is either blue or red. A player topples one of his/her dominoes to the left or to the right. David Wolfe asks if all dyadic rationals occur as a unique single row of dominoes and if that row is always palindromic (symmetrical). D6. Hanoi Stick-up is played with the disks of the Towers of Hanoi puzzle, starting with each disk in a separate stack. A move is to place one stack on top of another such that the size of the bottom of the ﬁrst stack is less than the size of the top of the second; the two stacks then fuse (&) into one. The only relevant information about a stack are its top and bottom sizes, and it’s often possible to collapse the labelling of positions: so for instance, starting with 8 disks and fusing 1&7 and 2&5

we have stacks 0 1&7 2&5 3 4 6 which can be relabelled 0 1&3 1&2 1 2 3 in which the legal moves are still the same. John Conway, Alex Fink and others have found that the -positions of height 3 in normal Hanoi Stickup are exactly those which, afterP collapsing, are of the≤ form 0a 01b 1c 12d 2e with min(a + b + c, c + d + e, a + e) even, except that if a + e a + b + c and a + e c + d + e then both a and e must be even (02 can’t≤ be involved in a legal move≤ so can be dropped). They also found the normal and mis`ere outcomes of all positions with up to six stacks, but there is more to be discovered.

D7(56). Are there any draws in Beggar-my-Neighbor ? Marc Paulhus showed that there are no cycles when using a half-deck of two suits, but the problem for the whole deck (one of Conway’s “anti-Hilbert” problems) is still open.

E. Theory of Games

E1(49). Fraenkel updates Berlekamp’s earlier questions on computational complexity as follows: Demaine, Demaine & Eppstein [2002] proved that deciding whether a player can win in a single move in Phutball (WW, pp. 752–755; LIP, p. 212) is NP- complete. Grossman & Nowakowski [2002] gave constructive partial strategies

22 for 1-dimensional Phutball. Thus, these papers do not show that Phutball has the required properties. Perhaps Nimania (Fraenkel & Neˇsetˇril [1985]) and Multivision (Fraenkel [1998]) satisfy the requirements. Nimania begins with a single positive integer, but after a while there is a multiset of positive integers on the table. At move k, a copy of an existing integer m is selected, and 1 is subtracted from it. If m = 1, the copy is deleted. Otherwise, k copies of m 1 are adjoined to the copy m 1. The player first unable to move loses and− the opponent wins. It − was proved: (i) The game terminates. (ii) Player I can win. In Fraenkel, Loebl & Neˇsetˇril [1988], it was shown that the max number of moves in Nimania is n 2 n 1 an Ackermann function, and the min number satisfies 22 − Min(n) 22 − . ≤ ≤ The game is thus intractable simply because of the length of its play. This is a provable intractability, much stronger than NP-hardness, which is normally only a conditional intractability. One of the requirements for the tractability of a game is that a winner can consummate a win in at most O(cn) moves, where c>1 is a constant, and n a sufficiently succinct encoding of the input (this much is needed for nim on 2 equal heaps of size n). To consummate a win in Nimania, player I can play randomly most of the time, but near the end of play, a winning strategy is needed, given explicitly. Whether or not this is an “intricate” solution, depends on the beholder. But it seems that it’s of even greater interest to construct a game with a very simple strategy which still has high complexity! Also every play of Multivision terminates, the winner can be determined in linear time, and the winning moves can be computed linearly. But the length of play can be arbitrarily long. So let’s ask the following: Is there a game which has

1. simple, playable rules,

2. a simple explicit strategy,

3. length of play at most exponential; and

4. is NP-hard or harder.

Tung [1987] proved the following: Theorem. Given a polynomial P (x, y) Z[x, y], the problem of deciding ∈ whether x y[P (x, y) = 0] holds over Z 0, is co-NP-complete. ∀ ∃ ≥

23 Deﬁne the following game of length 2: player I picks x Z 0, player II ∈ ≥ picks y Z 0. Player I wins if P (x, y) = 0, otherwise player II wins. For ∈ ≥ , winning, player II has only to compute y such that P (x, y) = 0, given x, and there are many algorithms for doing so. Also Jones & Fraenkel [1995] produced games, with small length of play, which satisfy these conditions. So we are led to the following reformulation of Berlekamp’s question: Is there a game which has

1. simple, playable rules,

2. a ﬁnite set of options at every move,

3. a simple explicit strategy,

4. length of play at most exponential;

5. and is NP-hard or harder.

E2. Complexity closure. Aviezri Fraenkel asks: Are there partizan games G1, G2, G3 such that: (i) G1, G2, G3, G1 + G2, G2 + G3 and all their options have polynomial-time strategies, (ii) G1 + G3 is NP-hard?

E3. Sums of switch games. David Wolfe considers a sum of games G, each of the form a b c or a b c where a, b, and c are integers speciﬁed in unary. Is there a polynomial2 | time| 2 algorithm to determine who wins in G, or is the problem NP-hard?

E4(52). How does one play sums of games with varied overheating oper- ators? Sentestrat and Top-down thermography (LIP, p. 214): David Wolfe would like to see a formal proof that sentestrat works, an algorithm for top-down thermography, and conditions for which top-down thermography is computationally eﬃcient. Aaron Siegel asks the following generalized thermography questions. (1) Show that the Left scaﬀold of a dogmatic (neutral ko-threat environment; LIP, p. 215) thermograph is decreasing as function of t. (Note, this is

24 NOT true for komaster thermographs.) [Dogmatic thermography was invented by Berlekamp & Spight. See [19] for a good introduction.] (2) Develop the machinery for computing dogmatic thermographs of double kos (multiple alternating 2-cycles joined at a single node). In the same vein as (2): (3) Develop a temperature theory that applies to all loopy games. Siegel thinks that (3) is among the most important open problems in combinatorial game theory. The temperature theory of Go appears radically different from the classical combinatorial theory of loopy games (where infinite plays are draws). It would be a huge step forward if these could be reconciled into a “grand unified temperature theory”. Problem (2) seems to be the obvious next step toward (3). Conway asks for a natural set of conditions under which the mapping G 5→ ∗ G is the unique homomorphism that annihilates all infinitesimals. 4 E5. Loopy games (WW, pp. 334–377; LIP, pp. 213–214) are partizan games that do not satisfy the ending condition. A stopper is a game that, when played on its own, has no ultimately alternating, Left and Right, infinite sequence of legal moves. Aaron Siegel reminds us of WW 2nd ed. p.369, where the authors tried hard to prove that every loopy game had stoppers, until Clive Bach found the Carousel counterexample. Is there an alternative notion of simplest form that works for all finite loopy games, and, in particular, for the Carousel? The simplest form theorem for stoppers is at WW, p.351. Siegel conjectures that, if Q is an arbitrary cycle of Left and Right moves that contains at least two moves for each player, and is not strictly alternating, then there is a stopper consisting of a single cycle that matches Q, together with various exits to enders, i.e., games which end in a finite, though possibly unbounded, number of moves. [Note that games normally have Left and Right playing alternately, but if the game is a sum, then play in one component can have abitrary sequences of Left and Right moves, not just alternating ones.] A long cycle is tame if it alternates just once between Left and Right, otherwise it is wild. Aaron Siegel writes:

I can produce wild cycles “in the laboratory,” by specifying their game graphs explicitly. So the question is to detect one “in nature”, i.e., in an actual game with (reasonably) playable rules such as Phutball [Problem B7].

25 Siegel also asks under what conditions does a given infinitesimal have a well-defined atomic weight, and asks to specify an algorithm to calculate the atomic weight of an infinitesimal stopper g. The algorithm should succeed whenever the atomic weight is well-defined, i.e., whenever g can be sandwiched between loopfree all-smalls of equal atomic weight.

E6(45). Elwyn Berlekamp asks for the habitat of 2, where 2= 0, 0, . Gabriel Drummond-Cole [2005] has found Domineering∗ positions∗ with{ ∗| this∗} value. See, for example, Figure 7, which also shows a Go position, found by Nakamura & Berlekamp [2003], whose chilled value is 2. The Black and ∗ White groups are both connected to life via unshown connections emanating upwards from the second row. Either player can move to by placing a stone at E2, or to 0 by going to E1. ∗

"" 3 " 2 "" 1 ABCDEFGH Figure 6: A Domineering position and a Chilled Go position of value 2 ∗

E7. Partial ordering of games. David Wolfe lets g(n) be the number of games born by day n, notes that an upper bound is given by g(n + 1) ≤α g(n)+2g(n) + 2, and a lower bound for each α< 0 is given by g(n+ 1) 2g(n) , for n suﬃciently large, and asks us to tighten these bounds. ≥ He also asks what group is generated by the all-small games (or—much harder—of all games) born by day 3. Describe the partial order of games born by day 3, identifying all the largest “hypercubes” (Boolean sublattices) and how they are interconnected. These questions have been answered for day 2, see this volume, pp.??.

Berlekamp suggests other possible definitions for games born by day n, n, depending on how one defines . Our definition is 0-based, as = 0 . OtherG G0 G0 { } natural definitions are integer-based (where are integers) or number-based. G0 These two alternatives do not form a lattice, for if G1 and G2 are born by day k, then the games

H := G ,G G , G n n 1 2 1 { 2|− } 5 & 6 & 26& form a decreasing sequence of games born by day k + 2 exceeding any game G G ,G , and the day k + 2 join of G and G cannot exist. What is the ≥ 1 2 1 2 structure of the partial order given by one of these alternative definitions of birthday? The set of all short games does not form a lattice, but Calistrate, Paulhus & Wolfe [2002] have shown that the games born by day n form a distributive lattice n under the usual partial order. They ask for a description of the exact structureL of . Siegel describes as “truly gigantic and exceedingly difficult L3 L4 to penetrate” but suggests that it may be possible to find its dimension and the maximum longitude, long (G), of a game in , which he defines as 4 L4

long (G) = rank (G G•) rank (G) n n ∨ − n where rank (G) is the rank of G in and G• is the companion of G, n Ln ifG =0 L ∗ R 0, (G )• (G )• if G>0 G• =  { L | R }  (G )• 0, (G )• if G<0  { L | R }  (G )• (G )• if G 0 { | } 2  The set of all-small games does not form a lattice, but Siegel forms a lattice 0 n by adjoining least and greatest elements and and asks: do the elements ofL 0 have an intrinsic “handedness” that7 distinguishes,8 say, (n 1) from Ln − ·↑ (n 1) + ? − ·↑ ∗ E8. Aaron Siegel asks, given a group or monoid, , of games, to specify K a technique for calculating the simplest game in each -equivalence class. He notes that some restriction on might be needed; forK example, might be K K the monoid of games absorbed by a given idempotent.

E9. Siegel also would like to investigate how search methods might be integrated with a canonical-form engine.

E10(9). Develop a mis`ere theory for unions of partizan games (WW, p. 312).

E11. Four-outcome-games. Guy has given a brute force analysis of a parity subtraction game [9] which didn’t allow the use of Sprague-Grundy theory because it was’t impartial, nor the Conway theory, because it was not

27 last-player-winning. Is there a class of games in which there are four outcomes, ext, revious, eft and ight, and for which a general theory can be given? N P L R Acknowledgement. We have had help in compiling this collection from all those mentioned, and from others. We would especially like to mention Elwyn Berlekamp, Aviezri Fraenkel, Thane Plambeck, Aaron Siegel & David Wolfe. All mistakes are deliberate and designed to keep the reader alert.

References

[those not listed here may be found in Fraenkel’s Bibliography] [1] M. H. Albert, J. P. Grossman, S. McCurdy, R. J. Nowakowski & D. Wolfe, Cherries, preprint, 2005. [Problem A5]

[2] Sa´ul A. Blanco & Aviezri S. Fraenkel, Tromineering, Tridomineering and L- Tridomineering, August 2006 preprint. [Problem B11]

[3] N. Bullock, Domineering: Solving large combinatorial search spaces, ICGA J., 25(2002) 67-85; also MSc thesis, Univ. of Alberta, 2002. [Problem B11]

[4] N. Comeau, J. Cullis, R. J. Nowakowski & J. Paek, personal communication (class project). [Problem A12]

[5] Paul Erd˝os & George Szekeres, A combinatorial problem in geometry, Compo- sitio Math., 2(1935) 464–470; Zbl 12 270–271. [Problem C7]

[6] P. A. Grillet, Commutative Semigroups, Advances in Mathematics, 2, Springer 2001. [Problem A15]

[7] J. P. Grossman & R. J. Nowakowski, A ruler regularity in hexadecimal games, preprint 2005. [Problem A3]

[8] Vladimir Gurvich, On the mis`ere version of game Euclid and miserable games, Discrete Math., (to appear). [Problem A9]

[9] Richard Guy, A parity subtraction game, Crux Math., 33(2007) (to appear) [Problem E11]

[10] Sarah McCurdy, Two Combinatorial Games, MSc thesis, Dalhousie Univ., 2004. [Problem A5]

[11] M. M¨uller. Solving 5 5 Amazons. In The 6th Game Programming Workshop × 2001, 14 in IPSJ Symposium Series Vol.2001, pp. 64-71, Hakone, Japan, 2001. [Problem B6]

28 [12] Ivars Peterson, Chaotic Chomp, the mathematics of crystal growth sheds light on a tantalizing game, Science News, 170 (2006-07-22) 58-60. [Problem D1]

[13] Thane E. Plambeck & Aaron N. Siegel, Mis`ere quotients of impartial games, J. Combin. Theory, Ser. A (submitted). [Problem A15]

[14] L. R´edei, The Theory of Finitely Generated Commutative Semigroups, Perga- mon, 1965. [Problem A15]

[15] Thea van Roode, Partizan Forms of Hackenbush, MSc. thesis, The University of Calgary, 2002. [Problem C5]

[16] C. Schensted, Longest increasing and decreasing subsequences, Canad. J. Math., 13(1961) 179–191; MR 22 #12047. [Problem C7]

[17] Aaron Nathan Siegel, Loopy Games and Computation, PhD dissertation, Univ. of California, Berkeley, Spring 2005. [passim]

[18] Angela Siegel, Finite excluded subtraction sets and Inﬁnite Geography, MSc thesis, Dalhousie Univ., 2005. [Problem A1]

[19] W. L. Spight, Evaluating kos in a neutral threat environment: Preliminary results. In J. Schaeffer, M. Müller & Y. Björnsson, editors, Computers and Games: Third Internat. Conf., CG’02, Lect. Notes Comput. Sci., 2883 Springer, Berlinn, 2003, pp.413–428. [Problem E4]

Richard K. Guy Department of Mathematics and Statistics The University of Calgary Calgary, Alberta Canada T2N 1N4 [email protected] Richard J. Nowakowski Department of Mathematics and Statistics Dalhousie University Halifax, NS Canada B3H 3J5 [email protected]